Speaker: Detlev Hoffmann, University of Nottingham

Title:  Levels of Quaternion Algebras

Abstract:
The level of a ring $R$ with $1\neq 0$ is the smallest positive
integer $s$ such that $-1$ can be written as a sum of $s$ squares
in $R$, provided $-1$ is a sum of squares at all. Otherwise, the level
is defined to be infinite.

A famous result by Pfister (1965) states that the level of a field,
if finite, is always a $2$-power, and that each $2$-power can be realized
as level of a suitable field.  This answered a question asked by
Van der Waerden in the 1930s.

In contrast, every positive integer can be realized as level of an
integral domain as was shown by Dai, Lam and Peng (1980).

D.W. Lewis (1987) showed that any value of type $2^n$ or
$2^n+1$ can be realized as level of a quaternion algebra, and he
asked whether there exist quaternion algebras whose levels are not of
that form. We give a positive answer to that question by
constructing examples using methods from the theory of function fields
of quadratic forms.